the Rabi frequency for the specific detuning parameter. The eigenstates associated with the energy eigenvalues are given by where the angle is defined through Schrödinger picture dynamics It is now possible to obtain the dynamics of a general state by expanding it on to the noted eigenstates. We consider a superposition of number states as the initial state for the field, , and assume an atom in the excited state is injected into the field. The initial state of the system is Since the are stationary states of the field-atom system, then the state vector for times is just given by The Rabi oscillations can readily be seen in the sin and cos functions in the state vector. Different periods occur for different number states of photons. What is observed in experiment is the sum of many periodic functions that can be very widely oscillating and destructively sum to zero at some moment of time, but will be non-zero again at later moments. Finiteness of this moment results just from discreteness of the periodicity arguments. If the field amplitude were continuous, the revival would have never happened at finite time. Heisenberg picture dynamics It is possible in the Heisenberg notation to directly determine the unitary evolution operator from the Hamiltonian: where the operator is defined as and is given by The unitarity of is guaranteed by the identities and their Hermitian conjugates. By the unitary evolution operator one can calculate the time evolution of the state of the system described by its density matrix , and from there the expectation value of any observable, given the initial state: The initial state of the system is denoted by and is an operator denoting the observable. Mathematical formulation 2 For ease of illustration, consider the interaction of two energy sub-levels of an atom with a quantized electromagnetic field. The behavior of any other two-state system coupled to a bosonic field will be isomorphic to these dynamics. In that case, the Hamiltonian for the atom-field system is: Where we have made the following definitions: is the Hamiltonian of the atom, where the letters are used to denote the excited and ground state respectively. Setting the zero of energy to the ground state energy of the atom simplifies this to where is the resonance frequency of transitions between the sub-levels of the atom. is the Hamiltonian of the quantized electromagnetic field. Note the infinite